f ANALYTICAL MECHANICS
where Ki is the radius of gyration of the element of mass: about the F'-axis. Integrating the last equation we have
Ki2dm+ I x2dm. JQ
Each of the elements of mass has its own F'-axis similarly placed. Therefore &i is the same for all the elements of mass. and remains constant during the integration. Hence
Xm x2 dm. It is not difficult to see ,
that Ii is the sum of the moments of inertia of all the elements of mass relative to their respective F'-axes. It is equal, therefore, to the moment of inertia about the F-axis of the lamina (A in the figure) which would be obtained if the entire cylinder could be compressed into the transverse section through the F-axis. On the other hand h equals the moment of inertia about the F-axis of the lamina (B in the figure) which would be obtained if the cylinder could be compressed into the longitudinal section through the F-axis.
As an illustration of the last theorem consider the illustrative example of p. 157.
Applying the theorem we see that the moment of inertia of the cylinder equals the sum of the moments of inertia of the two lamina of Fig. 9L